## Periods of cusp forms and elliptic curves over imaginary quadratic fields

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- by J. E. Cremona and E. Whitley PDF
- Math. Comp.
**62**(1994), 407-429 Request permission

## Abstract:

In this paper we explore the arithmetic correspondence between, on the one hand, (isogeny classes of) elliptic curves*E*defined over an imaginary quadratic field

*K*of class number one, and on the other hand, rational newforms

*F*of weight two for the congruence subgroups ${\Gamma _0}(\mathfrak {n})$, where

*n*is an ideal in the ring of integers

*R*of

*K*. This continues work of the first author and forms part of the Ph.D. thesis of the second author. In each case we compute numerically the value of the

*L*-series $L(F,s)$ at $s = 1$ and compare with the value of $L(E,1)$ which is predicted by the Birch-Swinnerton-Dyer conjecture, finding agreement to several decimal places. In particular, we find that $L(F,1) = 0$ whenever $E(K)$ has a point of infinite order. Several examples are given in detail from the extensive tables computed by the authors.

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## Additional Information

- © Copyright 1994 American Mathematical Society
- Journal: Math. Comp.
**62**(1994), 407-429 - MSC: Primary 11F67; Secondary 11F66, 11G05, 11G40
- DOI: https://doi.org/10.1090/S0025-5718-1994-1185241-6
- MathSciNet review: 1185241